AP Physics C E&M- 12.3 Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law- Study Notes- New Syllabus

Magnetic Field Produced by a Current-Carrying Wire

A moving charge or current produces a magnetic field in the surrounding space. The direction of the magnetic field around a straight current-carrying wire is determined by the right-hand rule: if the thumb points in the direction of current, the curled fingers show the circular magnetic field lines.

Biot–Savart Law (General Expression):

The magnetic field at a point due to a small current element is given by:

\( \delta \vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{I \, \delta \vec{l} \times \hat{r}}{r^2} \)

• \(\mu_0\): permeability of free space \((4\pi \times 10^{-7} \, \mathrm{T \cdot m/A})\).
• \(I\): current in the wire.
• \(\delta \vec{l}\): small element of the wire in the direction of current.
• \(\hat{r}\): unit vector pointing from the current element to the field point.
• \(r\): distance between the current element and the field point.

Magnetic Field Produced by a Current-Carrying Wire

An electric current consists of moving charges, and these moving charges produce a magnetic field around the wire. The strength and direction of this magnetic field depend on the current and the geometry of the wire.

Right-Hand Rule:

• Point the thumb of your right hand in the direction of the conventional current (\(I\)).
• The curl of your fingers shows the circular direction of the magnetic field lines around the wire.

Magnetic Field Around a Long Straight Wire:

Using Ampère’s law or the Biot–Savart law, the magnetic field at a perpendicular distance \(r\) from a long straight wire carrying current \(I\) is:

\( \mathrm{B = \dfrac{\mu_0 I}{2 \pi r}} \)

• \(\mathrm{\mu_0 = 4\pi \times 10^{-7} \, T \cdot m/A}\) is the permeability of free space.
• The field decreases with distance (\(1/r\)).
• Field lines are concentric circles around the wire.

Magnetic Field Inside a Current-Carrying Wire:

For a wire of radius \(R\), carrying uniform current distribution:

\( \mathrm{B(r) = \dfrac{\mu_0 I r}{2 \pi R^2}}, \quad (r < R) \)

• Inside the wire, the field increases linearly with distance from the axis.
• At the surface (\(r = R\)), this matches the external formula.

Magnetic Field Due to a Circular Current Loop:

At the center of a circular loop of radius \(R\) carrying current \(I\):

\( \mathrm{B = \dfrac{\mu_0 I}{2R}} \)

• Direction given by the right-hand rule (thumb = current, curl = field).

Properties:

• The field strength increases with current (\(I\)) and decreases with distance (\(r\)).
• Magnetic field lines form closed loops around the current.
• The principle of superposition applies: fields from multiple wires add vectorially.